In statistics, a sequence or a vector of random variables is homoscedastic ( /ËhoĘmoĘskÉËdĂŚstÉŞk/) if all random variables in the sequence or vector have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The spellings homoskedasticity and heteroskedasticity are also used frequently.
The assumption of homoscedasticity simplifies mathematical and computational treatment. Serious violations in homoscedasticity (assuming a distribution of data is homoscedastic when in actuality it is heteroscedastic (/ËhÉtÉroĘskÉËdĂŚstÉŞk/)) may result in overestimating the goodness of fit as measured by the Pearson coefficient.
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As used in describing simple linear regression analysis, one assumption of the fitted model (to ensure that the least-squares estimators are each a best linear unbiased estimator of the respective population parameters, by the GaussâMarkov theorem) is that the standard deviations of the error terms are constant and do not depend on the x-value. Consequently, each probability distribution for y (response variable) has the same standard deviation regardless of the x-value (predictor). In short, this assumption is homoscedasticity. Homoscedasticity is not required for the estimates to be unbiased, consistent, and asymptotically normal.
Residuals can be tested for homoscedasticity using the BreuschâPagan test, which regresses square residuals to independent variables. Since the BreuschâPagan test is sensitive to normality, the KoenkerâBasset or 'generalized BreuschâPagan' test is used for general purposes. Testing for groupwise heteroscedasticity requires the GoldfeldâQuandt test.
Two or more normal distributions, , are homoscedastic if they share a common covariance (or correlation) matrix, . Homoscedastic distributions are especially useful to derive statistical pattern recognition and machine learning algorithms. One popular example is Fisher's linear discriminant analysis.
The concept of homoscedasticity can be applied to distributions on spheres.[1]
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